A Genuine Rank--dependent dependent Generalization of von

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A G i R k

A G i R k d d d d A Genuine Rank

A Genuine Rank--dependent dependent Generalization of von

Generalization of von Generalization of von Generalization of von Neumann

Neumann--Morgenstern Morgenstern Neumann

Neumann Morgenstern Morgenstern Expected Utility

Expected Utility p p y y

Mohammed Abdellaoui CNRS

CNRS France

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Outline Outline

Ou e

I. Review and Motivation

II. Probability Tradeoffs

III Probability Tradeoff Consistency

III. Probability Tradeoff Consistency

IV. Cumulative Prospect Theoryp y

V. Concluding Remarks

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I. Review and Motivation I. Review and Motivation.. ev ewev ew dd o vo v oo

E i t l Fi di

Experimental Findings

P & B (1948) AJP

Preston & Baratta (1948), AJP

The authors explored whether subjects accounted for chance events at their true mathematical probabilities.p

Edwards (1954) PR

Edwards (1954), PR

The author observed that subject’s bets

l d f b bili i

revealed preferences among probabilities.

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Th Fi t M d l The First Models

Handa (1977), JPE

( _i ) ( )_i

i w p u x

∑

ﬂ Violation of First Order Stochastic

i

Dominance

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K h & T k (1979) Kahneman & Tversky (1979) Prospect Theory

Prospect Theory

The Fourfold Patern of Risk Attitudes cannot be The Fourfold Patern of Risk Attitudes cannot be explained by the utility function for money

GAINS LOSSES

L Pr b bilit Ri k kin Ri k r i n Low Probability Risk seeking Risk aversion High Probability Risk aversion Risk seeking

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Q i i (1982) Quiggin (1982)

Rank-dependent EU Theory

An enhanced version of prospect theory avoiding violations of FSD.

It takes into account an additional behavioral rule:

The attention given to an outcome depends not g p only on the probability of the outcome but also on the favorability of the outcome in

comparison to the other outcomes .

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Ill i

Illustration

Assume that a decision maker is a pessimist and evaluates the lottery

(30$, 1/3 ; 20$, 1/3 ; 10$, 1/3).

( )

The DM will pay more attention (π₃) than 1/3 to the worst outcome 10. Say that the decision weight for outcome 10 is outcome 10. Say that the decision weight for outcome 10 is π₃=1/2.

The DM accordingly pays relatively less attention to the The DM, accordingly, pays relatively less attention to the other outcomes (π₁₊π₂=1/2). Being a pessimist, he will pay more than helf of the remaining attention to outcome 20; say

1/3 Th i d f h g i d d h b y

π₂=1/3. The remainder of the attention, devoted to the best outcome, is small (1/6).

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The Common Probabilit Weighting F nction

1 • Tversky & Kahneman

The Common Probability Weighting Function

1 Tversky & Kahneman

(1992), JRU

w(p) •Wu & Gonzalez (1996),

Management Science

•Bleichrodt & Pinto (2000)

0 1

(2000)

•Abdellaoui (2000) Management Science p

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E i ti A i ti ti f RDEU Existing Axiomatizations of RDEU

• Rank-dependent Expected Utility

1 1 1 1

( : ;...; : ) ...

( ) ( )

n n n n

P p x p x with x x x

RDU P

= −

∑

í í í

1

( ) ( )

( ... ) ( ... )

i i i

i n i n i

RDU P u x

w p p w p p

π

π ₋

=

= + + − + +

∑

• Outcome-oriented Axiomatizations of RDEU:

( ) ( 1)

i pn pi pn pi₋

Outcome-oriented Axiomatizations of RDEU:

Quiggin (1982), Segal (1989), Chew (1989), Wakker (1994), Chateauneuf (1995), Nakamura (1995)

• No obvious link with the von Neumann Morgenstern axiomatic set-up.

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RDEU d h All i P d

RDEU and the Allais Paradox

0.01

0.89

1 M

0.01

0.89

0 M

1 M

0.10 1 M 0.10

5 M

0.01 0.89

1 M

0.01 0.89

0 M

0 M 1 M^{0 M}

0.89 0.10

1 M

0.89 0.10

1 M

5 M

0 M ^{0 M}

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II. Probability Tradeoffs II. Probability Tradeoffs.. ob bob b yy deo sdeo s

P li i i Preliminaries

Measuring Utility from Mixtures

M is a mixture set

[0,1], x y, M : x (1 ) y M

α α α

∀ ∈ ∀ ∈ + − ∈

x, y, z are three consequences in M such that

[0,1], x y, M : x (1 ) y M

α α α

∀ ∈ ∀ ∈ + ∈

If Then …

x f fy z

1 1

2 2

y f x + z

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M i U ili f T d ff

Measuring Utility from Tradeoffs

Let D = X² a set of alternatives and É a preference relation.

~ : indifference and Ä : strict preference

Coordinate 2 Coordinate 2

a'

Coordinate 1 a

δ γ β α

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Revealed Tradeoffs Revealed Tradeoffs

( , ) ~ ( , ')

[ ] ~ [ ]

( ) ( ')

def t

a a

γ δ

αβ γδ β

⎧ ⎫

⎨ ⎬ ⇒

⎩ ⎭ ] ]

( , ) ~ ( , ')a a γ

α β

⎨ ⎬

⎩ ⎭

( , ) ( , ')

[ ] [ ]

def t

a a

γ δ

αβ γδ

⎧ ⎫

⎨ Ç ⎬ ⇒ f

[ ] [ ]

( , )a ( , ')a αβ γδ

α β ⇒

⎨ ⎬

⎩ ⎭ f

í

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II.

II. Probability.. Probability Tradeoffsob bob b yy Tradeoffsdeo sdeo s

1

...

1

x í x í í x

n

1

...

1

n n

x í x

₋

í í x

*

n

i j

p =

∑

p

Probability of receiving xⁱ or any

j j i=

∑

Probability of receiving xⁱ or any better outcome.

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P b bili /R k d d T i l Probability/Rank-ordered Triangle

* * *

3 2 1 ( 1, 3) ( 2 , 3 )

x í x í x P = p p P = p p

1 1

p^* p 3

3

P P*

3 2 1 ( 1, 3) ( 2 , 3 )

x í x í x P p p P p p

1 1

Increa sing

P Preferences

P_X P*_X

p2 g Prefere

nces Increasing Pr

*

p1

p^*₂ p₂

¤ ¤

P^* P

p3 p^*

3

p^*₂

0 p 1 0 1

1

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II.

DEFINITION DEFINITION

For probabilities α, β, γ, δ we write [αβ] t For probabilities α, β, γ, δ we write [αβ] t [γ δ] if

( P* i) (β Q* i) d (α, P*-i) (β, Q*-i) and (γ, P*-i) (δ, Q*-i)

(γ, ) ( , Q )

for some rank-ordered set {x1, …, xn} and

I {2 } h th t i i 1 d P*

I {2, …, n} such that xi xi-1 and P*, Q* .

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D i d P b bili T d ff Derived Probability Tradeoffs

p* 3

1

3

• OBSERVATION:

Under EU we have

[ β] ^t [ δ]

α β

P

Q

*

O O

[αβ] ~ [^t γδ]

⇓

p2

*

γ δ

R

S

*

O O

α β γ δ

⇓

=

0 p q 1

2* 2* α β γ δ− = −

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II.

Allais Parado 0 1M 5M

Allais Paradox: x1 = 0, x2 = 1M, x3 = 5M Alternatives in P Alternatives in P*

Problem 1 ^P⁼ (0, 1, 0)

Q = (0.01, 0.89, 0.1)

P*=(1,0) Q* = (0.99, 0.10)

Problem 2 ^R = (0.89, 0.11, 0) S = (0.90, 0,0.10)

R* = (0.11, 0.10) S* = (0.10, 0.10)

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All i P d ( i !) Allais Paradox (once again!)

p^*₃

1

p₃

• CONCLUSION:

[1 0 99] ^t [0 11 0 10]

[1; 0.99] f^t [0.11; 0.10]

p2

*

* Q S

easuring rod' 0.10

O O

0

p2

* P*

e'm R

0.10 0.11 0.99 1

O O

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O h E i l Fi di

Other Experimental Findings

1 0.8 0.9 Abdellaoui & Munier (1998)

5 0.6 0.7 3 0.4 0.50.1 0.2 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0

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III. Tradeoff Consistency III. Tradeoff Consistency.. deodeo Co s s e cyCo s s e cy

R k d d E d U ili

Rank-dependent Expected Utility

* *

1 2 1 2 3 2 3

( ) [ ] [ ]

EU P = +u u − u p + u − u p

↓

* *

1 2 1 2 3 2 3

( ) [ ] ( ) [ ] ( )

RDU P u u u w p u u w p

↓

= + − + −

LEMMA: Under RDEU

[αβ] ~ [^t γδ] ⇒ w( )α − w( )β = w( )γ − w( )δ [αβ] f^t [γδ] ⇒ w( )α − w( )β > w( )γ − w( )δ

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P b bili T d ff C i

Probability Tradeoff Consistency

p₃^* 1

*

α β

P

R

Q

*

P'

R'

Q'

*

* O

O O

O

0 1

p₂^* γ

δ

R

S ^*

m m'

R

S'^*

O O

m m

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MAIN THEOREM MAIN THEOREM

Let be a preference relation on . Then the following two statements are equivalent:

(i) RDU holds on ;

( ) ;

(ii) The following conditions are satisfied

i k rd r n ;

a. is a weak order on ; b. satisfies FSD;

c. is Jensen continuous;

d. satisfies tradeoff consistencyy^.

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III.

III. Tradeoff.. Tradeoff Consistencydeodeo ConsistencyCo s s e cyCo s s e cy

PROPOSITION PROPOSITION

Let be a vNM-independent weak order on . p Then:

(i) satisfies FSD;

(ii) [α β ] t [γ δ ] w(α) – w(β) ≥ w(γ) – w(δ);

(iii)[ β ] [ δ ] ( ) (β) ( ) (δ) (iii)[α β ] t [γ δ ] w(α) – w(β) > w(γ) – w(δ);

(iv) [α β ] ~t [γ δ ] w(α) – w(β) = w(γ) – w(δ);

(v) satisfies tradeoff consistency.

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III.

III. TradeoffConsistency.. TradeoffConsistencydeo Co s s e cydeo Co s s e cy

COROLLARY ( NM Th 1944) COROLLARY (vNM Theorem, 1944)

Let be a preference relation on . Expected p p Utility holds if and only if:

(i) is a weak order;

(ii) i J i

(ii) is Jensen continuous;

(iii) is vNM-independent.

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IV. Cumulative Prospect Theory IV. Cumulative Prospect TheoryV. Cu uV. Cu u veve ospecospec eo yeo y

U ili d CPT (Di i i hi i i i ) Utility under CPT (Diminishing sensitivity)

UTILITY

GAINS

LOSSES GAINS

LOSSES

0

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IV. Cumulative Prospect Theory IV. Cumulative Prospect TheoryV. Cu uV. Cu u veve ospecospec eo yeo y

U ili f i f i d l

- Utility function for gains and losses;

- Utility is concave for gains and - Utility is concave for gains and convex for losses with u(0)=0;

- Utility steeper forlosses than for gains (near 0);

gains (near 0);

- Two probability weighting p y g g functions: gainsand losses.

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IV. Cumulative Prospect Theory IV. Cumulative Prospect TheoryV. Cu uV. Cu u veve ospecospec eo yeo y

x₁ ≥ x₂ ≥ 0 # w⁺(p)U(x₁) + (1-w⁺(p))U(x₂)

p

1 2

Gains

x₁

p

0 ≥ x₂ ≥ x₁ # w^-(p)U(x₁) + (1-w^-(p))U(x₂)

x₂

1-p ^Losses

x₁₁ > 0 > x₂₂ # w⁺(p) ((p)U(x₁₁) + (w) ( ( p)) (^-(1-p))U(x₂₂)) Mixed

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w

inverse-S, p

expected utility

ational

extreme inverse-S ("fifty-fifty")

motivam

pessimism prevailing

finding pessimistic fif fif

cognitive

p finding fifty-fifty

Abdellaoui (2000); Bleichrodt & Pinto (2000); Gonzalez &

Wu 1999; Tversky & Fox, 1997.

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IV. Probabilistic Risk Attitude IV. Probabilistic Risk AttitudeV.V. ob b s cob b s c ss udeude

P A f P b bili i Ri k

« Pratt-Arrow » for Probabilistic Risk THEOREM

Suppose that RDEU holds for _i , w_i, u_i, i 1,2. Then the following two statements are equivalent :

(i) w₂ w₁ for a continuous, convex (respectively concave), strictly increasing : 0,1 0,1 ;

(ii) i ( i l ) b bili i i k

(ii) ₂ is more averse (respectively prone) to probabilistic risk than ₁ .

Definition

exhibitsprobabilistic risk aversionif ☺; ^t ☺ ; exhibitsprobabilistic risk aversionif ☺; ☺ ;

is excluded for all probabilities considered with☺ , 0.

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IV. Probabilistic Risk Attitude IV. Probabilistic Risk AttitudeV.V. ob b s cob b s c ss udeude

COROLLARY COROLLARY

Under RDEU wis convex (concave linear) if and only if Under RDEU,wis convex (concave, linear) if and only if

exhibits probabilistic riskaversion (proneness, both aversion and proneness)

aversion and proneness).

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V. Concluding Remarks V. Concluding Remarks V. Co c ud g e s V. Co c ud g e s

C ib i

Contributions:

This paper provides the first genuine p p p g

generalization of the vNM EU theorem.

It t hniq r imm di t l dir t d Its techniques are immediately directed towards the non-linear processing of

probabilities.

Its techniques allow for straightforward Its techniques allow for straightforward testing and elicitation of RDEU.